Odd-Rule Cellular Automata on the Square Grid

TL;DR

This paper classifies all odd-rule cellular automata on a square grid with neighborhoods up to size 9, identifying 86 unique behaviors and highlighting the most rapidly expanding rule that activates nearly 75% of cells.

Contribution

It provides a complete classification of 86 odd-rule cellular automata based on neighborhood subsets and their growth sequences, using a meta-algorithm to derive generating functions.

Findings

Identified 86 distinct odd-rule CAs modulo symmetries.

Reduced to 48 unique growth sequences after removing duplicates.

Found the fastest-growing CA activates nearly 75% of cells.

Abstract

An "odd-rule" cellular automaton (CA) is defined by specifying a neighborhood for each cell, with the rule that a cell turns ON if it is in the neighborhood of an odd number of ON cells at the previous generation, and otherwise turns OFF. We classify all the odd-rule CAs defined by neighborhoods which are subsets of a 3 X 3 grid of square cells. There are 86 different CAs modulo trivial symmetries. When we consider only the different sequences giving the number of ON cells after n generations, the number drops to 48, two of which are the Moore and von Neumann CAs. This classification is carried out by using the "meta-algorithm" described in an earlier paper to derive the generating functions for the 86 sequences, and then removing duplicates. The fastest-growing of these CAs is neither the Fredkin nor von Neumann neighborhood, but instead is one defined by "Odd-rule" 365, which turns ON almost 75% of all possible cells.